Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  trlnle Unicode version

Theorem trlnle 30680
Description: The atom not under the fiducial co-atom  W is not less than the trace of a lattice translation. Part of proof of Lemma C in [Crawley] p. 112. (Contributed by NM, 26-May-2012.)
Hypotheses
Ref Expression
trlne.l  |-  .<_  =  ( le `  K )
trlne.a  |-  A  =  ( Atoms `  K )
trlne.h  |-  H  =  ( LHyp `  K
)
trlne.t  |-  T  =  ( ( LTrn `  K
) `  W )
trlne.r  |-  R  =  ( ( trL `  K
) `  W )
Assertion
Ref Expression
trlnle  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  -.  P  .<_  ( R `  F
) )

Proof of Theorem trlnle
StepHypRef Expression
1 simpl1l 1008 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( F `
 P )  =  P )  ->  K  e.  HL )
2 hlatl 29855 . . . . 5  |-  ( K  e.  HL  ->  K  e.  AtLat )
31, 2syl 16 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( F `
 P )  =  P )  ->  K  e.  AtLat )
4 simpl3l 1012 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( F `
 P )  =  P )  ->  P  e.  A )
5 trlne.l . . . . 5  |-  .<_  =  ( le `  K )
6 eqid 2412 . . . . 5  |-  ( 0.
`  K )  =  ( 0. `  K
)
7 trlne.a . . . . 5  |-  A  =  ( Atoms `  K )
85, 6, 7atnle0 29804 . . . 4  |-  ( ( K  e.  AtLat  /\  P  e.  A )  ->  -.  P  .<_  ( 0. `  K ) )
93, 4, 8syl2anc 643 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( F `
 P )  =  P )  ->  -.  P  .<_  ( 0. `  K ) )
10 simpl1 960 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( F `
 P )  =  P )  ->  ( K  e.  HL  /\  W  e.  H ) )
11 simpl3 962 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( F `
 P )  =  P )  ->  ( P  e.  A  /\  -.  P  .<_  W ) )
12 simpl2 961 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( F `
 P )  =  P )  ->  F  e.  T )
13 simpr 448 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( F `
 P )  =  P )  ->  ( F `  P )  =  P )
14 trlne.h . . . . . 6  |-  H  =  ( LHyp `  K
)
15 trlne.t . . . . . 6  |-  T  =  ( ( LTrn `  K
) `  W )
16 trlne.r . . . . . 6  |-  R  =  ( ( trL `  K
) `  W )
175, 6, 7, 14, 15, 16trl0 30664 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( F  e.  T  /\  ( F `  P )  =  P ) )  ->  ( R `  F )  =  ( 0. `  K ) )
1810, 11, 12, 13, 17syl112anc 1188 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( F `
 P )  =  P )  ->  ( R `  F )  =  ( 0. `  K ) )
1918breq2d 4192 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( F `
 P )  =  P )  ->  ( P  .<_  ( R `  F )  <->  P  .<_  ( 0. `  K ) ) )
209, 19mtbird 293 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( F `
 P )  =  P )  ->  -.  P  .<_  ( R `  F ) )
215, 7, 14, 15, 16trlne 30679 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  P  =/=  ( R `  F ) )
2221adantr 452 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( F `
 P )  =/= 
P )  ->  P  =/=  ( R `  F
) )
23 simpl1l 1008 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( F `
 P )  =/= 
P )  ->  K  e.  HL )
2423, 2syl 16 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( F `
 P )  =/= 
P )  ->  K  e.  AtLat )
25 simpl3l 1012 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( F `
 P )  =/= 
P )  ->  P  e.  A )
26 simpl1 960 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( F `
 P )  =/= 
P )  ->  ( K  e.  HL  /\  W  e.  H ) )
27 simpl3 962 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( F `
 P )  =/= 
P )  ->  ( P  e.  A  /\  -.  P  .<_  W ) )
28 simpl2 961 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( F `
 P )  =/= 
P )  ->  F  e.  T )
29 simpr 448 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( F `
 P )  =/= 
P )  ->  ( F `  P )  =/=  P )
305, 7, 14, 15, 16trlat 30663 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( F  e.  T  /\  ( F `  P )  =/=  P ) )  ->  ( R `  F )  e.  A
)
3126, 27, 28, 29, 30syl112anc 1188 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( F `
 P )  =/= 
P )  ->  ( R `  F )  e.  A )
325, 7atncmp 29807 . . . 4  |-  ( ( K  e.  AtLat  /\  P  e.  A  /\  ( R `  F )  e.  A )  ->  ( -.  P  .<_  ( R `
 F )  <->  P  =/=  ( R `  F ) ) )
3324, 25, 31, 32syl3anc 1184 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( F `
 P )  =/= 
P )  ->  ( -.  P  .<_  ( R `
 F )  <->  P  =/=  ( R `  F ) ) )
3422, 33mpbird 224 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( F `
 P )  =/= 
P )  ->  -.  P  .<_  ( R `  F ) )
3520, 34pm2.61dane 2653 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  -.  P  .<_  ( R `  F
) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721    =/= wne 2575   class class class wbr 4180   ` cfv 5421   lecple 13499   0.cp0 14429   Atomscatm 29758   AtLatcal 29759   HLchlt 29845   LHypclh 30478   LTrncltrn 30595   trLctrl 30652
This theorem is referenced by:  cdlemc3  30687
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-rep 4288  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371  ax-un 4668
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-nel 2578  df-ral 2679  df-rex 2680  df-reu 2681  df-rab 2683  df-v 2926  df-sbc 3130  df-csb 3220  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-op 3791  df-uni 3984  df-iun 4063  df-br 4181  df-opab 4235  df-mpt 4236  df-id 4466  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5385  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-ov 6051  df-oprab 6052  df-mpt2 6053  df-1st 6316  df-2nd 6317  df-undef 6510  df-riota 6516  df-map 6987  df-poset 14366  df-plt 14378  df-lub 14394  df-glb 14395  df-join 14396  df-meet 14397  df-p0 14431  df-p1 14432  df-lat 14438  df-clat 14500  df-oposet 29671  df-ol 29673  df-oml 29674  df-covers 29761  df-ats 29762  df-atl 29793  df-cvlat 29817  df-hlat 29846  df-lhyp 30482  df-laut 30483  df-ldil 30598  df-ltrn 30599  df-trl 30653
  Copyright terms: Public domain W3C validator